Fault identifying method for sludge bulking based on a recurrent RBF neural network

ABSTRACT

The wastewater treatment process by using activated sludge process often appear the sludge bulking fault phenomenon. Due to production conditions of wastewater treatment process, the correlation and restriction between variables, the characteristics of nonlinear and time-varying, which lead to hard identification of sludge bulking; Sludge bulking is not easy to detect and the reasons resulting in the sludge bulking are difficult to identify, are current RBF neural network is designed for detecting and identifying the causes of sludge volume index (SVI) in this patent. The method builds soft-computing model of SVI based on recurrent RBF neural network, it has been completed to the real-time prediction of SVI concentration and better accuracy were obtained. Once the fault of sludge bulking is detected, the identifying cause variables (CVI) algorithm can find the cause variables of sludge bulking. The method can effectively identify the fault of sludge bulking and ensure the safety operation of the wastewater treatment process.

CROSS REFERENCE TO RELATED PATENT APPLICATIONS

This application claims priority to Chinese Patent Application No. 201710186738.5, filed on Mar. 27, 2017, entitled “an fault identifying method for sludge bulking based on a recurrent RBF neural network,” which is hereby incorporated by reference in its entirety.

TECHNICAL FIELD

In this present disclosure, an identifying method is designed for detecting the sludge bulking and find the fault reasons of sludge bulking in the urban wastewater treatment process (WWTP) by a recurrent RBF neural network. The sludge volume index-SVI is an important parameter of characterization of sludge bulking. The basic link of proposed identifying method is to predict SVI on the basis of the relationships between variables. The technology of this present disclosure is part of advanced manufacturing technology, belongs to both the field of control engineer and environment engineer. Therefore, the identifying method for detecting the sludge bulking and find the fault reasons of sludge bulking in WWTP is of great significance.

BACKGROUND

The urban WWTP also happen the sludge bulking in the activated sludge process. However, the various influencing factors for SVI are various and complex. Therefore, it is difficult to identify the cause variables for sludge bulking, which seriously affected the stable operation of the urban WWTP. The identifying method for sludge bulking, based on recurrent RBF neural network, is helpful to detect the fault phenomenon of sludge bulking and identify the cause variables that resulted in the sludge bulking, which strengthen delicacy management, ensure water quality effluent standards of urban WWTP. It has better economic benefit as well as significant environmental and social benefits. Thus, the research achievements have wide application prospect in this present disclosure.

At present, the activated sludge process has widely used in the urban wastewater treatment process of papermaking, printing and dyeing, chemical industry, and many other industrial wastewater. However, sludge bulking fault problem has always been a thorny problems existing in the activated sludge system. Due to the sludge and water can't normal separate, cause the failure of wastewater treatment process. Sludge bulking occurs frequently and it has basically different degrees of sludge bulking in the wastewater treatment process. High sludge bulking coverage, in many countries such as Germany, Britain, South Africa's wastewater treatment plants according to the survey, more than half of the wastewater treatment plants exist the situation of the excessive growth of filamentous bacteria. Thus, sludge bulking is a common problem in wastewater treatment plants at home and abroad. Many scholars of most countries have studied the prevention and control method of the sludge bulking. Although some progress has been made, but so far, there is no effective control measures of sludge bulking; Moreover, in the event of the sludge bulking, the reason is not easy to explore, and need longer time to treat the failure of wastewater treatment. To sum up, once in the event of sludge bulking, consequences cannot be ignored. For this failure phenomenon of sludge bulking, therefore, early diagnosis and prevention is the most effective methods to solve the problem of sludge bulking, so it has high practical significance.

In this present disclosure, an identifying method for sludge bulking, is presented by building a soft-computing model based on recurrent RBF neural network. The neural network uses fast gradient descent algorithm to ensure the accuracy of recurrent RBF neural network. Once the sludge bulking is detected, an identifying cause variables (CVI) algorithm will be exploited to implement the identification of fault variables. This method can effectively prevent the happen of sludge bulking and reduce the loss of the wastewater treatment plant.

SUMMARY

A fault identification method is designed for the sludge bulking based on a recurrent RBF neural network. Its characteristic and steps include following steps:

(1) Determine the Input and Output Variables of SVI:

For sewage treatment process of activated sludge system, by analyzing the detailed mechanism of sludge bulking, five process variables are analyzed and select the input variables of SVI soft-computing model: dissolved oxygen concentration-DO, mixed liquor suspended solids concentration-MLSS, temperature-T, chemical oxygen demand-COD and total nitrogen-TN. The output value of soft-computing model is detected SVI concentration.

(2) Initial Recurrent RBF Neural Network:

The structure of recurrent RBF neural network comprise three layers: input layer, hidden layer and output layer. The network is 5-J−1, named the number of input layer is 5 and hidden neurons is J. Connection weights between input layer and hidden layer are assigned 1, the connection weights between hidden layer and output layer randomly assign values, the assignment interal is [1, 1]. The number of the training sample is N and the input of recurrent RBF neural network is x(t)=[x₁(t), x₂(t), x₃(t), x₄(t), x₅(t)] at time t. The expectations output of neural network output is expressed as y_(d)(t) and the actual output is expressed as y(t). Soft-computing method of SVI can be described:

{circle around (1)} The input Layer: There are 5 neurons which represent the input variables in this layer. The output values of each neuron are as follows: u _(i)(t)=x _(i)(t)  (1) wherein u_(i)(t) is the ith output value at time t, i=1, 2, . . . , 5, and the input vector is x(t)=[x₁(t), x₂(t), . . . , x₅(t)].

{circle around (2)} The Hidden Layer: There are J neurons of hidden layer. The outputs of hidden neurons are:

$\begin{matrix} {{{\theta_{j}(t)} = e^{- \frac{{{{h_{j}{(t)}} - {c_{j}{(t)}}}}^{2}}{2\;{\sigma_{j}^{2}{(t)}}}}},{j = 1},2,\ldots\mspace{14mu},J} & (2) \end{matrix}$ c_(j)(t) denotes the center vector of the jth hidden neuron and c_(j)(t)=[c_(j1)(t), c_(j2)(t), . . . , c_(jn+1)(t)]^(T) at time t, ∥h_(j)(t)−c_(j)(t)∥ is the Euclidean distance between h_(j)(t) and c_(j)(t), and σ_(j)(t) is the radius or width of the jth hidden neuron at time t, h_(j)(t) is input vector of the jth hidden neuron at time t described as h _(j)(t)=[u ₁(t),u ₂(t), . . . u ₅(t),v _(j)(t)×y(t−1)]^(T)  (3) y(t−1) is the output value of the output layer at time t−1, 1), v_(j)(t) denotes the connection weight from output layer to the jth hidden neuron at time t, and v(t)=[v₁(t), v₂(t), . . . , v_(J)(t)]^(T), T represents transpose.

{circle around (3)} The Output Layer: There is only one node in this layer, the output is:

$\begin{matrix} {{{y(t)} = {{f\left( {{w(t)},{\theta(t)}} \right)} = {\sum\limits_{j = 1}^{J}\;{{w_{j}(t)} \times {\theta_{j}(t)}}}}},{j = 1},L,J} & (4) \end{matrix}$ wherein w(t)=[w₁(t), w₂(t), . . . , w_(J)(t)]^(T) is the connection weights between the hidden neurons and output neuron at time t, θ(t)=[θ₁(t), θ₂(t), . . . , θ_(J)(t)]^(T) is the output vector of the hidden layer, y(t) represents the output of recurrent RBF neural network at time t.

The error of recurrent RBF neural network is:

$\begin{matrix} {{E(t)} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\;\left( {{y_{d}(t)} - {y(t)}} \right)^{2}}}} & (5) \end{matrix}$ y_(d)(t) is the expectation output of neural network and the actual output is expressed as y(t).

(3) Train Recurrent RBF Neural Network:

{circle around (1)} Given the recurrent RBF neural network, the initial number of hidden layer neurons is J, J>2 is a positive integer. The input of recurrent RBF neural network is x(1), x(2), . . . , x(t), . . . , x(N), correspondingly, the expectation output is y_(d)(1), y_(d)(2), . . . , y_(d)(t), . . . , y_(d)(N), expected error value is set to E_(d), E_(d)∈(0, 0.01). The every variable of initial centre value c₁(1)∈(−2, 2), width value σ_(j)(1)∈(0, 1), initial feedback weight v_(j)(1)∈(0, 1), j=1, 2, . . . , J. Initial weight w(1)∈(0, 1).

{circle around (2)} Set the learning step s=1;

{circle around (3)} t=s; According to Eqs. (1)-(4), calculate the output of recurrent RBF neural network, exploiting fast gradient descent algorithm:

$\begin{matrix} {\mspace{79mu}{{c_{j}\left( {t + 1} \right)} = {{c_{j}(t)} - {\eta_{c}\frac{1}{\sigma_{j}^{2}}\left( {{y_{d}(t)} - {y(t)}} \right){w_{j}(t)} \times {{\theta(t)}\left\lbrack {{h_{j}(t)} - {c_{j}(t)}} \right\rbrack}}}}} & (6) \\ {{\sigma_{j}\left( {t + 1} \right)} = {{\sigma_{j}(t)} - {\eta_{\sigma}\frac{1}{\sigma_{j}^{3}}\left( {{y_{d}(t)} - {y(t)}} \right){w_{j}(t)} \times {\theta(t)}{{{h_{j}(t)} - {c_{j}(t)}}}^{2}}}} & (7) \\ {\mspace{79mu}{{v_{j}\left( {t + 1} \right)} = {{v_{j}(t)} - {{\eta_{v}\left( {{y_{d}(t)} - {y(t)}} \right)}{w_{j}(t)}{\theta(t)}{y\left( {t - 1} \right)}}}}} & (8) \\ {\mspace{79mu}{{w_{j}\left( {t + 1} \right)} = {{w_{j}(t)} - {{\eta_{w}\left( {{y_{d}(t)} - {y(t)}} \right)}{\theta_{j}(t)}}}}} & (9) \end{matrix}$ η_(c), η_(σ), η_(v), η_(w) are the learning rate of centre, width, feedback connection weight from output layer to hidden layer and the connection weight between hidden layer and output layer, respectively. In addition, η_(c)∈(0, 0.01], η_(σ)∈(0, 0.01], η_(v) ∈(0, 0.02], η_(w)∈(0, 0.01]. c_(j)(t+1)=[c_(1j)(t+1), c_(2j) (t+1), . . . , c_(5j)(t+1)] denotes the center vector of the jth hidden neuron at time t+1. σ_(j)(t+1) is the radius or width of the jth hidden neuron at time t+1. v_(j)(t+1) denotes the connection weight from output layer to the jth hidden neuron at time t+1. w_(j)(t+1) is the connection weights between the hidden neurons and output neuron at time t+1.

{circle around (4)} increase 1 learning step s, if s<N, then turn to step {circle around (3)}. If s=N, turn to step {circle around (5)}.

{circle around (5)} according to Eq. (5), calculate the performance of recurrent RBF neural network. If E(t)≥E_(d), then turn to step {circle around (3)}; If E(t)<E_(d), stop the training process.

(4) SVI Concentration Prediction:

The testing samples used as the input of recurrent RBF neural network, the output of neural network is the soft-computing values of SVI.

(5) CVI Algorithm for Sludge Bulking:

{circle around (1)} Calculate the residual of the expectation output and the output of recurrent RBF neural network, if

$\begin{matrix} \left\{ \begin{matrix} {{{y(t)} - {y_{d}(t)}} \geq 5} \\ {{y(t)} \geq 150} \end{matrix} \right. & (10) \end{matrix}$ then turn to step {circle around (2)}, otherwise, stop the process of fault identification for sludge bulking.

{circle around (2)} Define two formula:

$\begin{matrix} \left\{ \begin{matrix} {{{IC}_{1}(t)} = {{\delta^{T}(t)}{\Lambda_{M}^{- 1}(t)}{\delta(t)}}} \\ {{{IC}_{2}(t)} = {{{\theta_{M}(t)}}^{2} - {{\delta^{T}(t)}{\delta(t)}}}} \end{matrix} \right. & (11) \end{matrix}$ wherein IC₁(t) is the Mahalanobis distance of input variables at time t and IC₂(t) is the squared prediction error at time t, M is the numbers of principal component of input data, θ_(M)(t) is the output vector of the hidden layer for M principal components training data at time t, K is the number of the front samples. δ(t)=[δ₁(t), . . . , δ_(m)(t), . . . , δ_(M)(y)]^(T) is the projection of the training data, and δ_(m)(t) is

$\begin{matrix} {{{\delta_{m}(t)} = {\sum\limits_{k = 1}^{K}{{a_{k}(t)}\left( {{{\overset{\_}{\theta}}_{m}(t)} \cdot {{\overset{\_}{\theta}}_{new}(t)}} \right)}}},{k = 1},L,{K;{m = 1}},\ldots\mspace{14mu},M} & (12) \end{matrix}$ wherein θ _(new)(t) is the mean-centered output vector of the hidden layer for M principal components training data at time t, θ _(m)(t) is the output vector of the hidden layer for the mth principal component training data at time t, a_(k)(t) is a constant, and a_(k)(t)∈(0, 0.01]. And the diagonal matrix of eigenvalues associated with M principal components is defined as

$\begin{matrix} {{\Lambda_{M}(t)} = {\begin{bmatrix} \lambda_{1} & \; & \; & \; \\ \; & \lambda_{2} & \; & \; \\ \; & \; & O & \; \\ \; & \; & \; & \lambda_{M} \end{bmatrix}\left( {\lambda_{1} \geq \lambda_{2} \geq L \geq \lambda_{M} \geq 0} \right)}} & (13) \end{matrix}$ wherein Λ_(M)(t) is the diagonal matrix of eigenvalues at time t and it satisfies {tilde over (C)}(t)=Z(t)Λ_(M)(t)Z ^(T)(t)+l′(t)(I(t)−Z(t)Z ^(T)(t))  (14) wherein l′(t) is a constant value, I(t) is a unit matrix, {tilde over (C)}(t) is the regularized covariance matrix of C(t) at time t:

$\begin{matrix} {{\Omega(t)} = {{\theta(t)}^{T}{{\overset{\sim}{C}}^{- 1}(t)}{\theta(t)}}} & (15) \\ {{\Omega(t)} = {{{IC}_{1}(t)} + {{l^{\prime - 1}(t)}{{IC}_{2}(t)}}}} & (16) \\ {{C(t)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{\theta_{k}(t)}{\theta_{k}(t)}^{T}}}}} & (17) \end{matrix}$ wherein Ω(t) is the energy of each variable. θ_(k)(t) is the hidden output vector of the kth principal component at time t.

The constant vector a(t)=[a₁(t), . . . a_(k)(t), . . . a_(K)(t)]^(T) is given as

$\begin{matrix} {{{\lambda(t)}{a(t)}} = {\frac{1}{K}{G(t)}{a(t)}}} & (18) \end{matrix}$ G(t) is the Gaussian matrix and λ(t) denotes the eigenvalue G(t)={θ_(i)(t)·θ_(j)(t)}_(K×K)  (19) λ(t)P(t)=C(t)p(t)  (20) p(t) denotes eigenvector of the covariance matrix C(t) at time t.

{circle around (3)} For the ith input variable, the contribution degree index satisfy:

$\begin{matrix} {{G^{i}(t)} = \frac{\kappa^{i}(t)}{\sum\limits_{i = 1}^{5}{\kappa^{i}(t)}}} & (21) \end{matrix}$ wherein G^(i)(t) is the contribution degree index of ith variable at time t, a hidden layer neuron corresponds to an input variable, κ^(i)(t) is the contribution degree which is calculate by the mutual information between this testing samples and difference sets for testing samples x^(i)(t) of ith variable at time t, which can be expressed as κ^(i)(t)=I(x ^(i)(t),V _(Δ)(t))  (22) I(x^(i)(t), V_(Δ)(t)) is the mutual information of x^(i)(t) and V_(Δ)(t) at time t, V_(Δ)(t) is the difference matrix sets of data of training set and testing set at time t, which is expressed as V _(Δ)(t)=V _(tr)(t)−V _(te)(t)  (23) wherein V_(tr)(t) and V_(te)(t) is independent data sets of training set and testing set at time t, respectively. V _(tr)(t)=D ⁻¹(t)G(t)  (24) V _(te)(t)=D _(te) ⁻¹(t)G _(te)(t)  (25) D(t) is the covariance matrix of Φ(t) at time t D(t)=E{Φ(t)Φ^(T)(t)}  (26) Φ(t)=[θ(t−K+1), . . . ,θ(t−1),θ(t−1),θ(t)]^(T)  (27) Φ(t) is output matrix of hidden layer at time t, θ(t−K+1) is the hidden output vector at time t−K+1, K is the number of the front samples.

{circle around (4)} For the ith input variable, a hidden layer neuron corresponds to an input variable, if the G^(i)(t) at time t satisfies: G ¹(t)+ . . . G ^(i)(t)≥0.8  (28)

the variables 1, . . . , i is the cause variables resulted in sludge bulking.

The Novelties of this Present Disclosure Contain:

(1) To detect the sludge bulking and identify the cause variables that resulted in the sludge bulking, an identifying method for sludge bulking is developed in this present disclosure. The results demonstrate that the SVI trends in WWTP can be predicted with acceptable accuracy using the DO, MLSS, T, TN and COD as input variables. This method can not only solve the problem of measured online for SVI concentration with acceptable accuracy but also detect the happen of sludge bulking.

(2) This identifying fault variables method is based on the CVI algorithm. And it identify the fault variables of sludge bulking in the WWTP with high identifying precision. Thus, it can realize the effective regulation of the sludge bulking control in advance.

Attention: this present disclosure utilizes five input variables in this identifying method to predict the SVI. In fact, it is in the scope of this present disclosure that any of the variables: DO, T, MLSS, COD and TN, are used to predict the SVI concentration. Moreover, this identifying method is also able to predict the others variables in urban WWTP.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the structure of identifying method based on the recurrent RBF neural network in this present disclosure.

FIG. 2 shows the testing result of the identifying method.

FIG. 3 shows the testing error of the identifying method.

FIG. 4 shows the IC₁(t) outputs of the identifying method.

FIG. 5 shows the IC₂(t) outputs of the identifying method.

FIG. 6 shows the marked fault points of IC₁(t) and IC₂(t) outputs of the identifying method.

FIG. 7 shows contribution degree index of input variables of identifying method.

DETAILED DESCRIPTION

This invention takes MLSS, DO, T, COD and TN as characteristic variables for SVI, the above unit is mg/L;

The experimental data comes from water quality analysis statement of a wastewater treatment plant in 2014; choosing data of MLSS, DO, T, COD and TN as experimental samples, after eliminating abnormal sample, 100 groups of data are available, and the group of 60 used as training samples, the remaining 40 groups as test samples.

This present disclosure adopts the following technical scheme and implementation steps:

A fault identification method is designed for the sludge bulking based on a recurrent RBF neural network. Its characteristic and steps include following steps:

(1) Determine the Input and Output Variables of SVI:

For sewage treatment process of activated sludge system, by analyzing the detailed mechanism of sludge bulking, five process variables are analyzed and select the input variables of SVI soft-computing model: dissolved oxygen concentration-DO, mixed liquor suspended solids concentration-MLSS, temperature-T, chemical oxygen demand-COD and total nitrogen-TN. The output value of soft-computing model is detected SVI concentration.

(2) Initial Recurrent RBF Neural Network:

The structure of recurrent RBF neural network comprise three layers: input layer, hidden layer and output layer in FIG. 1. The network is 5-5-1, named the number of input layer is 5 and hidden neurons is 5. Connection weights between input layer and hidden layer are assigned 1, the connection weights between hidden layer and output layer randomly assign values, the assignment interal is [1, 1]. The number of the training sample is N and the input of recurrent RBF neural network is x(t)=[x₁(t), x₂(t), x₃(t), x₄(t), x₅(t)] at time t. The expectations output of neural network output is expressed as y_(d)(t) and the actual output is expressed as y(t). Soft-computing method of SVI can be described:

{circle around (1)} The input Layer: There are 5 neurons which represent the input variables in this layer. The output values of each neuron are as follows: u _(i)(1)=x _(i)(t)  (29) wherein u_(i)(t) is the ith output value at time t, i=1, 2, . . . , 5, and the input vector is x(t)=[x₁(t), x₂(t), . . . , x₅(t)].

{circle around (2)} The Hidden Layer: There are J neurons of hidden layer. The outputs of hidden neurons are:

$\begin{matrix} {{{\theta_{j}(t)} = e^{- \frac{{{{h_{j}{(t)}} - {c_{j}{(t)}}}}^{2}}{2{\sigma_{j}^{2}{(t)}}}}},{j = 1},2,\ldots\mspace{14mu},J} & (30) \end{matrix}$ c_(j) (t) denotes the center vector of the jth hidden neuron and c_(j)(t)=[c_(j1)(t), c_(j2)(t), . . . , c_(jn+1)(t)]^(T) at time t, ∥h_(j)(t)−c_(j)(t)∥ is the Euclidean distance between h_(j)(t) and c_(j)(t), and σ_(j)(t) is the radius or width of the jth hidden neuron at time t, h_(j)(t) is input vector of the jth hidden neuron at time t described as h _(j)(t)=[u ₁(t),u ₂(t), . . . u ₅(t),v _(j)(t)×y(t−1)]^(T)  (31) y(t−1) is the output value of the output layer at time t−1, v_(j)(t) denotes the connection weight from output layer to the jth hidden neuron at time t, and v(t)=[v₁(t), v₂(t), . . . , v_(J)(t)]^(T), T represents transpose.

{circle around (3)} The Output Layer There is only one node in this layer, the output is:

$\begin{matrix} {{{y(t)} = {{f\left( {{w(t)},{\theta(t)}} \right)} = {\sum\limits_{j = 1}^{J}{{w_{j}(t)} \times {\theta_{j}(t)}}}}},{j = 1},\ldots\mspace{14mu},J} & (32) \end{matrix}$ wherein w(t)=[w₁(t), w₂(t), . . . , w_(J)(t)]^(T) is the connection weights between the hidden neurons and output neuron at time t, θ(t)=[θ₁(t), θ₂(t), . . . , θ_(J)(t)]^(T) is the output vector of the hidden layer, y(t) represents the output of recurrent RBF neural network at time t.

The error of recurrent RBF neural network is:

$\begin{matrix} {{E(t)} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\left( {{y_{d}(t)} - {y(t)}} \right)^{2}}}} & (33) \end{matrix}$

y_(d)(t) is the expectation output of neural network and the actual output is expressed as y(t).

(3) Train Recurrent RBF Neural Network:

{circle around (1)} Given the recurrent RBF neural network, the initial number of hidden layer neurons is J, J>2 is a positive integer. The input of recurrent RBF neural network is x(1), x(2), . . . , x(t), . . . , x(N), correspondingly, the expectation output is y_(d)(1), y_(d)(2), . . . , y_(d)(t), . . . , y_(d)(N), expected error value is set to E_(d), E_(d)∈(0, 0.01). The every variable of initial centre value c_(j)(1)∈(−2, 2), width value σ_(j)(1)∈(0, 1), initial feedback weight v_(j)(1)∈(0, 1), j=1, 2, . . . , J. Initial weight w(1)∈(0, 1).

{circle around (2)} Set the learning step s=1;

{circle around (3)} t=s; According to Eqs. (1)-(4), calculate the output of recurrent RBF neural network, exploiting fast gradient descent algorithm:

$\begin{matrix} {\mspace{79mu}{{c_{j}\left( {t + 1} \right)} = {{c_{j}(t)} - {\eta_{c}\frac{1}{\sigma_{j}^{2}}\left( {{y_{d}(t)} - {y(t)}} \right){w_{j}(t)} \times {{\theta(t)}\left\lbrack {{h_{j}(t)} - {c_{j}(t)}} \right\rbrack}}}}} & (34) \\ {{\sigma_{j}\left( {t + 1} \right)} = {{\sigma_{j}(t)} - {\eta_{\sigma}\frac{1}{\sigma_{j}^{3}}\left( {{y_{d}(t)} - {y(t)}} \right){w_{j}(t)} \times {\theta(t)}{{{h_{j}(t)} - {c_{j}(t)}}}^{2}}}} & (35) \\ {\mspace{79mu}{{v_{j}\left( {t + 1} \right)} = {{v_{j}(t)} - {{\eta_{v}\left( {{y_{d}(t)} - {y(t)}} \right)}{w_{j}(t)}{\theta(t)}{y\left( {t - 1} \right)}}}}} & (36) \\ {\mspace{79mu}{{w_{j}\left( {t + 1} \right)} = {{w_{j}(t)} - {{\eta_{w}\left( {{y_{d}(t)} - {y(t)}} \right)}{\theta_{j}(t)}}}}} & (37) \end{matrix}$ η_(c), η_(σ), η_(v), η_(w) are the learning rate of centre, width, feedback connection weight from output layer to hidden layer and the connection weight between hidden layer and output layer, respectively. In addition, η_(c)∈(0, 0.01], η_(σ)∈(0, 0.01], η_(v)∈(0, 0.02], η_(w)∈(0, 0.01]. c_(j)(t+1)=[c_(1j)(t+1), c_(2j)(t+1), . . . , c_(5j)(t+1)] denotes the center vector of the jth hidden neuron at time t+1. σ_(j)(t+1) is the radius or width of the jth hidden neuron at time t+1. v_(j)(t+1) denotes the connection weight from output layer to the jth hidden neuron at time t+1. w_(j)(t+1) is the connection weights between the hidden neurons and output neuron at time t+1.

{circle around (4)} increase 1 learning step s, if s<N, then turn to step {circle around (3)}. If s=N, turn to step {circle around (5)}.

{circle around (5)} according to Eq. (5), calculate the performance of recurrent RBF neural network. If E(t)≥E_(d), then turn to step {circle around (3)}; If E(t)<E_(d), stop the training process.

(4) SVI Concentration Prediction:

The testing samples used as the input of recurrent RBF neural network, the output of neural network is the soft-computing values of SVI. The testing result is shown in FIG. 2. X axis indicates the number of samples. Y axis shows SVI. The unit of Y axis is mg/L. The red solid line presents the real values of SVI. The blue dot line shows the outputs of recurrent RBF neural network in the testing process. The errors between the real values and the outputs of recurrent RBF neural network in the testing process are shown in FIG. 3. X axis shows the number of samples. Y axis shows the testing error. The unit of Y axis is mg/L.

(5) CVI Algorithm for Sludge Bulking:

{circle around (1)} Calculate the residual of the expectation output and the output of recurrent RBF neural network, if

$\begin{matrix} \left\{ \begin{matrix} {{{y(t)} - {y_{d}(t)}} \geq 5} \\ {{y(t)} \geq 150} \end{matrix} \right. & (38) \end{matrix}$ then turn to step {circle around (2)}, otherwise, stop the process of fault identification for sludge bulking.

{circle around (2)} Define two formula:

$\begin{matrix} \left\{ \begin{matrix} {{{IC}_{1}(t)} = {{\delta^{T}(t)}{\Lambda_{M}^{- 1}(t)}{\delta(t)}}} \\ {{{IC}_{2}(t)} = {{{\theta_{M}(t)}}^{2} - {{\delta^{T}(t)}{\delta(t)}}}} \end{matrix} \right. & (39) \end{matrix}$ wherein IC₁(t) is the Mahalanobis distance of input variables at time t and IC₂(t) is the squared prediction error at time t, M is the numbers of principal component of input data. K is the number of the front samples, θ_(M)(t) is the output vector of the hidden layer for M principal components training data at time t. δ(t)=[δ₁(t), . . . , δ_(m)(t), . . . , δ_(M)(t)]^(T) is the projection of the training data, and δ_(m)(t) is

$\begin{matrix} {{{\delta_{m}(t)} = {\sum\limits_{k = 1}^{K}{{a_{k}(t)}\left( {{{\overset{\_}{\theta}}_{m}(t)} \cdot {{\overset{\_}{\theta}}_{new}(t)}} \right)}}},{k = 1},L,{K;{m = 1}},\ldots\mspace{14mu},M} & (40) \end{matrix}$ wherein θ _(new)(t) is the mean-centered output vector of the hidden layer for M principal components training data at time t, θ _(m)(t) is the output vector of the hidden layer for the mth principal component training data at time t, K is the number of the front samples, a_(k)(t) is a constant, and a_(k)(t)∈(0, 0.01]. And the diagonal matrix of eigenvalues associated with M principal components is defined as

$\begin{matrix} {{\Lambda_{M}(t)} = {\begin{bmatrix} \lambda_{1} & \; & \; & \; \\ \; & \lambda_{2} & \; & \; \\ \; & \; & O & \; \\ \; & \; & \; & \lambda_{M} \end{bmatrix}\left( {\lambda_{1} \geq \lambda_{2} \geq L \geq \lambda_{M} \geq 0} \right)}} & (41) \end{matrix}$ wherein Λ_(M)(t) is the diagonal matrix of eigenvalues at time t and it satisfies {tilde over (C)}(t)=Z(t)Λ_(M)(t)Z ^(T)(t)+l′(t)(I(t)−Z(t)Z ^(t)(t))  (42) wherein l′(t) is a constant value, I(t) is a unit matrix, {tilde over (C)}(t) is the regularized covariance matrix of C(t) at time t:

$\begin{matrix} {{{\Omega(t)} = {{\theta(t)}^{T}\mspace{11mu}{C^{\% - 1}(t)}{\theta(t)}}}\;} & (43) \\ {{\Omega(t)} = {{{IC}_{1}(t)} + {{l^{\prime - 1}(t)}{{IC}_{2}(t)}}}} & (44) \\ {{C(t)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{\theta_{k}(t)}{\theta_{k}(t)}^{T}}}}} & (45) \end{matrix}$ wherein Ω(t) is the energy of each variable. θ_(k)(t) is the hidden output vector of the kth principal component at time t.

The constant vector a(t)=[a₁(t), . . . a_(j)(t), . . . a_(K)(t)]^(T) is given as

$\begin{matrix} {{{\lambda(t)}{a(t)}} = {\frac{1}{K}{G(t)}{a(t)}}} & (46) \end{matrix}$ G(t) is the Gaussian matrix and λ(t) denotes the eigenvalue G(t)={θ_(i)(t)·θ_(j)(t)}_(K×K)  (47) λ(t)p(t)=C(t)p(t)  (48) p(t) denotes eigenvector of the covariance matrix C(t) at time t.

{circle around (3)} For the ith input variable, the contribution degree index satisfy:

$\begin{matrix} {{G^{i}(t)} = \frac{\kappa^{i}(t)}{\sum\limits_{i = 1}^{5}{\kappa^{i}(t)}}} & (49) \end{matrix}$ wherein G^(i)(t) is the contribution degree index of ith variable at time t, a hidden layer neuron corresponds to an input variable, κ^(i)(t) is the contribution degree which is calculate by the mutual information between this testing samples and difference sets for testing samples x^(i)(t) of ith variable at time t, which can be expressed as κ^(i)(t)=I(x ^(i)(t),V _(Δ)(t))  (50) I(x^(i)(t), V_(Δ)(t)) is the mutual information of x^(i)(t) and V_(Δ)(t) at time t, V_(Δ)(t) is the difference matrix sets of data of training set and testing set at time t, which is expressed as V _(Δ)(t)=V _(tr)(t)−V _(te)(t)  (51) wherein V_(tr)(t) and V_(te)(t) is independent data sets of training set and testing set at time t, respectively. V _(tr)(t)=D ⁻¹(t)G(t)  (52) V _(te)(t)=D _(te) ⁻¹(t)G _(te)(t)  (53) D(t) is the covariance matrix of Φ(t) at time t D(t)=E{Φ(t)Φ^(T)(t)}  (54) Φ(t)=[(t−K+1), . . . ,θ(t−1),θ(t)]^(T)  (55) Φ(t) is output matrix of hidden layer at time t, θ(t−K+1) is the hidden output vector at time t−K+1, K is the number of the front samples.

{circle around (4)} For the ith input variable, a hidden layer neuron corresponds to an input variable, if the G^(i)(t) at time t satisfies: G ¹(t)+ . . . G ^(i)(t)≥0.8  (56)

the variables 1, . . . , i is the cause variables resulted in sludge bulking.

The IC₁(t) outputs is shown in FIG. 4. X axis shows the number of samples. Y axis shows the IC₁(t) outputs. The IC₂(t) outputs is shown in FIG. 5. X axis shows the number of samples. Y axis shows the IC₂(t) outputs. The marked fault points of IC₁(t) and IC₂(t) is shown in FIG. 6. X axis shows the number of samples. Y axis shows the marked fault points of IC₁(t) and IC₂(t). The green solid line shows the marked fault points of IC₁(t) in the testing process. The red solid line shows the marked fault points of IC₂(t) in the testing process. The contribution degree index of process variables is shown in FIG. 7, X axis shows the process variables. Y axis shows the contribution degree index.

Tables 1-12 show the experimental data in this present disclosure. Tables 1-5 show the training samples of COD, DO, T, MLSS and TN. Tables 6-10 show the testing samples of COD, DO, T, MLSS and TN. Table 11 shows real output values of SVI. Table 12 shows the outputs of the recurrent RBF neural network in the predicting process. Moreover, the samples are imported as the sequence from the tables. The first data is in the first row and the first column. Then, the second data is in the first row and the second column. Until all of data is imported from the first row, the data in the second row and following rows are inputted as the same way.

Training samples are provided as follows:

TABLE 1 The input of chemical oxygen demand-COD (mg/L) 317.655 319.9375 322.25 324.5625 326.875 329.1875 331.5 333.8125 336.125 338.4375 340.75 343.0345 347.6875 350 350.1875 350.375 350.5625 350.75 350.9375 351.125 351.3125 351.5 351.6875 351.875 352.0625 352.25 352.4375 352.625 352.8125 353 348.0625 343.125 338.1875 333.25 328.3125 323.375 318.4375 313.5 308.5625 303.625 305.6875 306.75 306.8125 307.875 308.9375 309 310.875 311.75 320.625 329.5 328.375 327.25 326.125 325 323.875 325.75 326.625 328.5 330.375 331.25

TABLE 2 The input of dissolved oxygen concentration-DO (mg/L) 6.845 7.246 6.659 7.239 6.255 6.735 6.481 6.724 6.944 7.434 6.917 7.842 6.128 7.901 7.405 7.647 7.879 7.434 7.179 7.234 6.532 6.543 6.554 6.088 6.974 6.222 7.893 6.058 6.664 7.753 6.695 6.38 7.751 6.112 6.935 7.038 7.506 6.355 6.152 6.222 7.974 6.129 6.853 7.138 6.178 6.555 6.617 7.151 7.924 6.525 6.899 7.194 7.555 7.29 7.044 6.975 6.311 7.014 6.556 6.108

TABLE 3 The input of T 24.1907 24.1592 24.1163 23.9890 23.9090 23.8377 23.7450 23.6837 23.6623 23.6367 23.6253 23.6295 23.6295 23.6239 23.6253 23.6310 23.6253 23.6224 23.6082 23.5968 23.5769 23.5470 23.5114 23.4745 23.4318 23.3835 23.3310 23.2770 23.2146 23.1451 23.0870 23.0403 23.0007 22.9625 22.9851 23.0191 23.0559 23.0743 23.1409 23.1877 23.2032 23.1565 23.1834 23.1707 23.1551 23.1409 23.0956 23.1027 23.0630 23.0134 22.9738 22.9115 22.8593 22.8098 22.7420 22.7448 22.7504 22.8084 22.9144 23.0191

TABLE 4 The input of mixed liquor suspended solids concentration-MLSS (mg/L) 1286.885 1287.375 1286.5 1285.625 1284.75 1283.875 1283 1282.125 1281.25 1283.375 1289.5 1288.625 1287.75 1287.875 1286 1287.313 1284.625 1283.938 1285.25 1282.563 1281.875 1281.188 1282.5 1283.813 1285.125 1284.438 1287.75 1287.063 1286.375 1285.688 1285 1285.188 1285.375 1287.563 1288.75 1289.938 1290.125 1296.313 1296.5 1296.688 1294.875 1297.063 1297.25 1297.438 1297.625 1297.813 1298 1297.375 1296.75 1296.125 1295.5 1293.875 1284.25 1293.625 1293 1302.375 1311.75 1321.125 1319.5 1329.875

TABLE 5 The input of total nitrogen-TN (mg/L) 41.275 41.3125 41.35 41.3875 41.425 42.4625 41.5 43.5375 41.575 44.6125 42.65 41.6875 41.725 44.7625 45.8 42.3125 42.825 43.3375 43.85 44.3625 44.875 45.3875 45.9 46.4125 46.925 47.4375 48.95 48.4625 48.975 49.4875 47.241 49.5375 49.075 48.6125 48.15 47.6875 47.225 46.7625 46.3 45.8375 47.375 48.9125 47.45 43.9875 43.525 44.0625 42.6 42.88125 43.1625 43.44375 43.725 44.00625 44.2875 44.56875 45.85 44.13125 47.4125 46.69375 45.975 46.25625 Testing samples:

TABLE 6 The input of chemical oxygen demand-COD (mg/L) 332.125 334 334.8125 335.625 336.4375 337.25 338.0625 338.875 339.6875 340.5 341.3125 342.125 342.9375 341.75 342.5625 343.375 344.1875 343 343.875 344.75 345.625 344.5 346.375 347.25 348.125 349 349.875 349.75 349.625 349.5 349.375 350.25 351.125 359 356.375 353.75 351.125 348.5 345.875 346.25

TABLE 7 The input of dissolved oxygen concentration-DO (mg/L) 6.59875 6.693125 6.7875 7.281875 7.17625 7.070625 6.965 6.859375 6.75375 6.648125 7.5425 6.936875 6.93125 6.725625 7.12 7.178125 6.83625 6.994375 6.9525 6.410625 6.46875 6.526875 7.085 6.643125 6.80125 6.659375 6.8175 6.875625 6.93375 6.991875 7.25 7.3375 7.125 7.1125 7 6.9875 6.975 6.9625 6.95 6.9375

TABLE 8 The input of T 23.1381 23.2841 23.4446 23.5598 23.6338 23.6680 23.6751 23.7307 23.7393 23.7535 23.7378 23.7193 23.6766 23.6295 23.6096 23.5570 23.4958 23.4489 23.4048 23.4006 23.3949 23.4048 23.4190 23.4915 23.8191 23.8477 23.8662 23.8905 23.0148 27.2087 27.2933 27.3334 27.317 27.3022 27.2888 27.2681 27.2354 27.1983 27.1584 27.0976

TABLE 9 The input of mixed liquor suspended solids concentration-MLSS (mg/L) 1329.25 1338.625 1348 1346.25 1344.5 1342.75 1345 1350.25 1357.5 1355.75 1354.09 1352.25 1350.5 1348.75 1347 1345.25 1343.5 1341.75 1340 1338.25 1336.5 1334.75 1333 1331.25 1329.5 1327.75 1326 1324.25 1322.5 1320.75 1319 1317.25 1315.5 1313.75 1312 1308.688 1305.375 1302.063 1298.75 1295.438

TABLE 10 The input of total nitrogen-TN (mg/L) 46.5375 46.81875 47.1 47.3875 49.675 47.9625 48.25 49.5375 50.825 49.1125 48.4 47.6875 46.975 45.2625 46.55 47.8375 46.125 47.4125 45.7 44.6375 44.575 43.5125 41.45 41.3875 41.325 41.2625 43.2 42.1375 40.075 40.0125 40.95 39.8875 40.825 39.7625 38.7 36.475 36.25 36.025 36.8 37.575

TABLE 11 The real output of sludge volume index-SVI (mg/L) 102.1318 104.5506 104.9427 105.2887 106.4809 107.2111 113.4479 119.0132 121.5167 115.047 119.9333 124.776 128.374 125.9713 130.4835 129.933 133.8057 135.4856 135.7096 134.7779 132.6632 136.047 135.7843 133.8904 131.9964 138.4294 144.1786 150.3707 152.1785 149.7829 154.45 154.7836 152.1172 157.832 155.5825 157.7072 158.4782 159.0304 160.8222 157.2703

TABLE 12 The output of sludge volume index-SVI in the testing process (mg/L) 104.6798 103.4415 106.0121 102.9707 104.3378 105.9182 112.4079 118.0338 122.0289 118.0726 122.6676 127.6083 133.1356 128.6487 133.2567 132.7159 137.5757 138.2312 140.2654 139.2904 137.7872 137.7449 141.9649 139.9922 136.1045 143.3388 147.961 155.0361 159.5865 157.078 159.0946 160.3922 161.725 164.3612 166.4679 166.6599 165.2393 164.7875 163.0705 162.5019 

What is claimed is:
 1. A fault identification method for sludge bulking based on a recurrent radial basis function neural network (RRBFNN) comprising the following steps: (1) determine input and output variables of sludge volume index (SVI): for sewage treatment process of an activated sludge system, by analyzing detailed mechanism of the sludge bulking, analyzing five process variables and selecting the input variables of SVI soft-computing model: dissolved oxygen concentration-DO, mixed liquor suspended solids concentration-MLSS, temperature-T, chemical oxygen demand-COD and total nitrogen-TN, wherein an output value of the SVI soft-computing model is SVI value; (2) initial RRBFNN: a structure of RRBFNN comprises three layers: an input layer, a hidden layer and an output layer, the RRBFNN adopts 5-J−1 connection, the number of neurons of the input layer is 5, the number of neurons of the hidden layer is J, where J is an integer larger than 2, and the number of neurons of the output layer is 1; connection weight between the input layer and the hidden layer is assigned 1, connection weight between the hidden layer and the output layer is randomly assigned a value in the range of [−1, 1]; the number of training samples is N and an input of the RRBFNN is x(t)=[x₁(t), x₂(t), x₃(t), x₄(t), x₅(t)] at time t; an expectation output of the RRBFNN is expressed as y_(d)(t) and an actual output of the RRBFNN is expressed as y(t); a soft-computing method of SVI is as follows: {circle around (1)} input layer: there are 5 neurons which represent the input variables in this layer, the output values of each neuron are as follows: u _(i)(t)=x _(i)(t)  (1) wherein u_(i)(t) is ith output value at time t, i=1, 2, . . . , 5; and an input vector is x(t)=[x₁(t), x₂(t), . . . , x_(n)(t)]; {circle around (2)} hidden layer: there are J neurons of the hidden layer, outputs of hidden neurons are: $\begin{matrix} {{{\theta_{j}(t)} = e^{- \frac{{{{h_{j}{(t)}} - {c_{j}{(t)}}}}^{2}}{2\;{\sigma_{j}^{2}{(t)}}}}},\mspace{14mu}{j = 1},2,K,J} & (2) \end{matrix}$ where θ_(j)(t) is an output value of jth hidden neuron at time t, e is an exponential function, c_(j)(t)=[c_(j1)(t), . . . , c_(j5)(t)]^(T) is a center vector of the jth hidden neuron at time t, ∥h_(j)(t)−c_(j)(t)∥ is Euclidean distance between h_(j)(t) and c_(j)(t), σ_(j)(t) is a width of the jth hidden neuron at time t, h_(j)(t) is input vector of the jth hidden neuron at time t described as h _(j)(1)=[u ₁(t),u ₂(t), . . . u ₅(t),v _(j)(t)×y(t−1)]^(T)  (3) where u₁(t) is dissolved oxygen concentration at time t, u₂(t) is mixed liquor suspended solids concentration at time t, u₃(t) is temperature at time t, u₄(t) is chemical oxygen demand at time t, u₅(t) is total nitrogen at time t, y(t−1) is an output value of an output neuron at time t−1, v_(j)(t) denotes the connection weight from th output layer to the jth hidden neuron at time t, and v(t)=[v₁(t), v₂(t), . . . , v_(J)(t)]^(T) is a vector of connection weight from the output laver to the hidden layer at time t, T represents transpose; {circle around (3)} output layer: there is only one node in this layer, the output is: $\begin{matrix} {{{y(t)} = {{f\left( {{w(t)},{\theta(t)}} \right)} = {\sum\limits_{j = 1}^{J}{{w_{j}(t)} \times {\theta_{j}(t)}}}}},\mspace{14mu}{j = 1},L,J} & (4) \end{matrix}$ where w(t)=[w₁(t), w₂(t), . . . , w_(J)(t)]^(T) is a vector of connection weight from the hidden layer to the output layer at time t, w_(j)(t) is connection weight from the jth hidden neuron to the output layer at time t, θ(t)=[θ₁(t), θ₂(t), . . . , θ_(J)(t)]^(T) is an output vector of the hidden layer, y(t) represents the output of RRBFNN at time t; the error of RRBFNN is: $\begin{matrix} {{E(t)} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\left( {{y_{d}(t)} - {y(t)}} \right)^{2}}}} & (5) \end{matrix}$ y_(d)(t) is an expectation output of the RRBFNN and an actual output is expressed as y(t); (3) train RRBFNN: {circle around (1)} given the RRBFNN, the input of RRBFNN is x(1), x(2), . . . , x(t), . . . , x(N), correspondingly, the expectation output is y_(d)(1), y_(d)(2), . . . , y_(d)(t), . . . , y_(d)(N), expected error value is set to E_(d), E_(d)∈(0, 0.01); initial centre value c_(j)(1)∈(−2, 2), width value σ_(j)(1)∈(0, 1), initial feedback weight v_(j)(1)∈(0, 1), j=1, 2, . . . , J; initial weight w(1)∈(0, 1); {circle around (2)} set the learning step s=1; {circle around (3)} t=s; according to Eqs. (1)-(4), calculate the output of RRBFNN, exploiting fast gradient descent algorithm: $\begin{matrix} {\mspace{79mu}{{c_{j}\left( {t + 1} \right)} = {{c_{j}(t)} - {\eta_{c}\frac{1}{\sigma_{j}^{2}}\left( {{y_{d}(t)} - {y(t)}} \right){w_{j}(t)} \times {{\theta(t)}\left\lbrack {{h_{j}(t)} - {c_{j}(t)}} \right\rbrack}}}}} & (6) \\ {{\sigma_{j}\left( {t + 1} \right)} = {{\sigma_{j}(t)} - {\eta_{\sigma}\frac{1}{\sigma_{j}^{3}}\left( {{y_{d}(t)} = {y(t)}} \right){w_{j}(t)} \times {\theta(t)}{{{h_{j}(t)} - {c_{j}(t)}}}^{2}}}} & (7) \\ {\mspace{79mu}{{v_{j}\left( {t + 1} \right)} = {{v_{j}(t)} - {{\eta_{v}\left( {{y_{d}(t)} - {y(t)}} \right)}{w_{j}(t)}{\theta(t)}{y\left( {t - 1} \right)}}}}} & (8) \\ {\mspace{79mu}{{w_{j}\left( {t + 1} \right)} = {{w_{j}(t)} - {{\eta_{w}\left( {{y_{d}(t)} - {y(t)}} \right)}{\theta_{j}(t)}}}}} & (9) \end{matrix}$ η_(c), η_(σ), η_(v), η_(w) are the learning rate of centre, width, feedback connection weight from output layer to hidden layer and the connection weight between hidden layer and output layer, respectively; in addition, η_(c)∈(0, 0.01], η_(σ)∈(0, 0.01], η_(v)E (0, 0.02], η_(w)∈(0, 0.01]; c_(j)(t+1)=[c_(1j)(t+1), c_(2j)(t+1), . . . , c_(5j)(t+1)] denotes the center vector of the jth hidden neuron at time t+1, σ_(j)(t+1) is the radius or width of the jth hidden neuron at time t+1, v_(j)(t+1) denotes the connection weight from output layer to the jth hidden neuron at time t+1, w_(j)(t+1) is the connection weights between the hidden neurons and output neuron at time t+1; {circle around (4)} increase 1 learning step s, if s<N, then turn to step {circle around (3)}; if s=N, turn to step {circle around (5)}; {circle around (5)} according to Eq. (5), calculate the performance of RRBFNN, if E(t)≥E_(d), then turn to step {circle around (3)}; if E(t)<E_(d), stop the training process; (4) SVI concentration prediction: testing samples used as the input of RRBFNN, the output of neural network is the soft-computing values of SVI; (5) cause variable identification (CVI) algorithm for sludge bulking: {circle around (1)} fault condition setting: calculate a residual of the expectation output and the output of RRBFNN at time t, if $\begin{matrix} \left\{ \begin{matrix} {{{y(t)} - {y_{d}(t)}} \geq 5} \\ {{y(t)} \geq 150} \end{matrix} \right. & (10) \end{matrix}$ then turn to step {circle around (2)}, otherwise, stop the process of fault identification for sludge bulking; {circle around (2)} fault point finding: define two formula: $\begin{matrix} \left\{ \begin{matrix} {{{IC}_{1}(t)} = {{\delta^{T}(t)}{\Lambda_{M}^{- 1}(t)}{\delta(t)}}} \\ {{{IC}_{2}(t)} = {{{\theta_{M}(t)}}^{2} - {{\delta^{T}(t)}{\delta(t)}}}} \end{matrix} \right. & (11) \end{matrix}$ wherein IC₁(t) is a Mahalanobis distance of input variables at time t and IC₂(t) is a squared prediction error at time t, M is numbers of principal component of input data; K is a number of front samples, δ(t)=[δ₁(t), . . . , δ_(i)(t), . . . , δ₅(t)]^(T) is a projection of the training data, and δ_(m)(t) is $\begin{matrix} {{{\delta_{m}(t)} = {\sum\limits_{k = 1}^{K}{{a_{k}(t)}\left( {{{\overset{\_}{\theta}}_{m}(t)} \cdot {{\overset{\_}{\theta}}_{new}(t)}} \right)}}},{k = 1},L,{K;{m = 1}},L,M} & (12) \end{matrix}$ wherein θ _(new)(t) is a mean-centered output vector of the hidden layer for M principal components training data at time t, θ _(m)(t) is the output vector of the hidden layer for the mth principal component training data at time t, a_(k)(t) is a constant, and a_(k)(t)∈(0, 0.01]; a diagonal matrix of eigenvalues associated with M principal components is defined as where λ₁(t), λ₂(t), λ₃(t), λ₄(t) and λ₅(t) are eigenvalues of training data matrix: Λ(t) satisfies $\begin{matrix} {{\Lambda_{M}(t)} = {\begin{bmatrix} \lambda_{1} & \; & \; & \; \\ \; & \lambda_{2} & \; & \; \\ \; & \; & O & \; \\ \; & \; & \; & \lambda_{M} \end{bmatrix}\left( {\lambda_{1} \geq \lambda_{2} \geq L \geq \lambda_{M} \geq 0} \right)}} & (13) \end{matrix}$ where λ₁, λ₂, . . . and λ_(M) are the eigenvalues of training data matrix, Λ_(M)(t) is the diagonal matrix of eigenvalues at time t and satisfies wherein l′(t) is a constant value, I(t) is a unit matrix, {tilde over (C)}(t) is a regularized covariance matrix of C(t) at time, C(t) is a mean square of hidden output vector: $\begin{matrix} {{{\Omega(t)} = {{\theta(t)}^{T}\mspace{11mu}{C^{\% - 1}(t)}{\theta(t)}}}\;} & (15) \\ {{\Omega(t)} = {{{IC}_{1}(t)} + {{l^{\prime - 1}(t)}{{IC}_{2}(t)}}}} & (16) \\ {{C(t)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{\theta_{k}(t)}{\theta_{k}(t)}^{T}}}}} & (17) \end{matrix}$ where Ω(t) is an energy of each variable, θ_(k)(t) is the hidden output vector of the kth principal component at time t; the constant vector a(t)=[a₁(t), . . . a_(j)(t), . . . a_(K)(t)]^(T) is given as $\begin{matrix} {{{\lambda(t)}{a(t)}} = {\frac{1}{K}{G(t)}{a(t)}}} & (18) \end{matrix}$ G(t) is a Gaussian matrix and λ(t) denotes an eigenvalue G(t)={θ_(i)(t)·θ_(j)(t)}_(K×K)  (19) λ(t)p(t)=C(t)p(t)  (20) p(t) denotes an eigenvector of a covariance matrix C(t) at time t; {circle around (3)} cause variable identification: for the ith input variable, a contribution degree index satisfies: $\begin{matrix} {{G^{i}(t)} = \frac{\kappa^{i}(t)}{\sum\limits_{i = 1}^{n}{\kappa^{i}(t)}}} & (21) \end{matrix}$ where G^(i)(t) is the contribution degree index of ith variable at time t, κ^(i)(t) is the contribution degree which is calculated by mutual information between this testing samples and difference sets for testing samples x^(i)(t) of ith variable at time t, which can be expressed as κ^(i)(t)=I(x ^(i)(t),V _(Δ)(t))  (22) I(x^(i)(t), V_(Δ)(t)) is the mutual information of x^(i)(t) and V_(Δ)(t) at time t, V_(Δ)(t) is difference matrix sets of data of training set and testing set at time t, which is expressed as V _(Δ)(t)=V _(tr)(t)−V _(te)(t)  (23) where V_(tr)(t) and V_(te)(t) is independent data sets of training set and testing set at time t, respectively: V _(tr)(t)=D ⁻¹(t)G(t)  (24) V _(te)(t)=D _(te) ⁻¹(t)G _(te)(t)  (25) D(t) is a covariance matrix of Φ(t) at time t, Φ(t) is output matrix of hidden layer at time t, D(t)=E{Φ(t)Φ^(T)(t)}  (26) Φ(t)=[θ(t−K+1), . . . ,θ(t−1),θ(t)]^(T)  (27) where E(Φ(t)) is the expectation of Φ(t), Φ(t) is output matrix of hidden layer at time t, θ(t−K+1) is the hidden output vector at time t−K+1; {circle around (4)} for the ith input variable, if the G^(i)(t) at time t satisfies: G ¹(t)+ . . . G ^(i)(t)≥0.8  (28) where variables 1, . . . , i are cause variables resulted in sludge bulking. 